# pidstd2

Create 2-DOF PID controller in standard form, convert to standard-form 2-DOF PID controller

## Syntax

C2 = pidstd2(Kp,Ti,Td,N,b,c)
C2 = pidstd2(Kp,Ti,Td,N,b,c,Ts)
C2 = pidstd2(sys)
C2 = pid2(___,Name,Value)

## Description

pid2 controller objects represent two-degree-of-freedom (2-DOF) PID controllers in parallel form. Use pid2 either to create a pid2 controller object from known coefficients or to convert a dynamic system model to a pid2 object.

Two-degree-of-freedom (2-DOF) PID controllers include setpoint weighting on the proportional and derivative terms. A 2-DOF PID controller is capable of fast disturbance rejection without significant increase of overshoot in setpoint tracking. 2-DOF PID controllers are also useful to mitigate the influence of changes in the reference signal on the control signal. The following illustration shows a typical control architecture using a 2-DOF PID controller.

C2 = pidstd2(Kp,Ti,Td,N,b,c) creates a continuous-time 2-DOF PID controller with proportional gain Kp, integrator and derivative time constants Ti, and Td, and derivative filter divisor N. The controller also has setpoint weighting b on the proportional term, and setpoint weighting c on the derivative term. The relationship between the 2-DOF controller’s output (u) and its two inputs (r and y) is given by:

$u={K}_{p}\left[\left(br-y\right)+\frac{1}{{T}_{i}s}\left(r-y\right)+\frac{{T}_{d}s}{\frac{{T}_{d}}{N}s+1}\left(cr-y\right)\right].$

This representation is in standard form. If all of the coefficients are real-valued, then the resulting C2 is a pidstd2 controller object. If one or more of these coefficients is tunable (realp or genmat), then C2 is a tunable generalized state-space (genss) model object.

C2 = pidstd2(Kp,Ti,Td,N,b,c,Ts) creates a discrete-time 2-DOF PID controller with sample time Ts. The relationship between the controller’s output and inputs is given by:

$u={K}_{p}\left[\left(br-y\right)+\frac{1}{{T}_{i}}IF\left(z\right)\left(r-y\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\left(cr-y\right)\right].$

IF(z) and DF(z) are the discrete integrator formulas for the integrator and derivative filter. By default,

$IF\left(z\right)=DF\left(z\right)=\frac{{T}_{s}}{z-1}.$

To choose different discrete integrator formulas, use the IFormula and DFormula properties. (See Properties for more information). If DFormula = 'ForwardEuler' (the default value) and N ≠ Inf, then Ts, Td, and N must satisfy Td/N > Ts/2. This requirement ensures a stable derivative filter pole.

C2 = pidstd2(sys) converts the dynamic system sys to a standard form pidstd2 controller object.

C2 = pid2(___,Name,Value) specifies additional properties as comma-separated pairs of Name,Value arguments.

## Input Arguments

 Kp Proportional gain. Kp can be: A real and finite value.An array of real and finite values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. Default: 1 Ti Integrator time. Ti can be: A real and positive value.An array of real and positive values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. When Ti = Inf, the controller has no integral action. Default: Inf Td Derivative time. Td can be: A real, finite, and nonnegative value.An array of real, finite, and nonnegative values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. When Td = 0, the controller has no derivative action. Default: 0 N Derivative filter divisor. N can be: A real and positive value.An array of real and positive values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. When N = Inf, the controller has no filter on the derivative action. Default: Inf b Setpoint weighting on proportional term. b can be: A real, nonnegative, and finite value.An array of real, nonnegative, finite values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. When b = 0, changes in setpoint do not feed directly into the proportional term. Default: 1 c Setpoint weighting on derivative term. c can be: A real, nonnegative, and finite value.An array of real, nonnegative, finite values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. When c = 0, changes in setpoint do not feed directly into the derivative term. Default: 1 Ts Sample time. To create a discrete-time pidstd2 controller, provide a positive real value (Ts > 0).pidstd2 does not support discrete-time controller with undetermined sample time (Ts = -1). Ts must be a scalar value. In an array of pidstd2 controllers, each controller must have the same Ts. Default: 0 (continuous time) sys SISO dynamic system to convert to standard pidstd2 form. sys be a two-input, one-output system. sys must represent a valid 2-DOF controller that can be written in standard form with Ti > 0, Td ≥ 0, and N > 0. sys can also be an array of SISO dynamic systems.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Use Name,Value syntax to set the numerical integration formulas IFormula and DFormula of a discrete-time pidstd2 controller, or to set other object properties such as InputName and OutputName. For information about available properties of pidstd2 controller objects, see Properties.

## Output Arguments

 C2 2-DOF PID controller, returned as a pidstd2 controller object, an array of pidstd2 controller objects, a genss object, or a genss array. If all the coefficients have scalar numeric values, then C2 is a pidstd2 controller object. If one or more coefficients is a numeric array, C2 is an array of pidstd2 controller objects. The controller type (such as PI, PID, or PDF) depends upon the values of the gains. For example, when Td = 0, but Kp and Ti are nonzero and finite, C2 is a PI controller. If one or more coefficients is a tunable parameter (realp), generalized matrix (genmat), or tunable gain surface (tunableSurface), then C2 is a generalized state-space model (genss).

## Properties

 b, c Setpoint weights on the proportional and derivative terms, respectively. b and c values are real, finite, and positive. When you create a 2-DOF PID controller using the pidstd2 command, the initial values of these properties are set by the b, and c input arguments, respectively. Kp Proportional gain. The value of Kp is real and finite. When you create a 2-DOF PID controller using the pidstd2 command, the initial value of this property is set by the Kp input argument. Ti Integrator time. Ti is real and positive. When you create a 2-DOF PID controller using the pidstd2 command, the initial value of this property is set by the Ti input argument. When Ti = Inf, the controller has no integral action. Td Derivative time. Td is real, finite, and nonnegative. When you create a 2-DOF PID controller using the pidstd2 command, the initial value of this property is set by the Td input argument. When Td = 0, the controller has no derivative action. N Derivative filter divisor. N must be real and positive. When you create a 2-DOF PID controller using the pidstd2 command, the initial value of this property is set by the N input argument. IFormula Discrete integrator formula IF(z) for the integrator of the discrete-time pidstd2 controller C2. The relationship between the inputs and output of C2 is given by: $u={K}_{p}\left[\left(br-y\right)+\frac{1}{{T}_{i}}IF\left(z\right)\left(r-y\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\left(cr-y\right)\right].$ IFormula can take the following values: 'ForwardEuler' — IF(z) = $\frac{{T}_{s}}{z-1}.$This formula is best for small sample time, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sample time, the ForwardEuler formula can result in instability, even when discretizing a system that is stable in continuous time.'BackwardEuler' — IF(z) = $\frac{{T}_{s}z}{z-1}.$An advantage of the BackwardEuler formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result.'Trapezoidal' — IF(z) = $\frac{{T}_{s}}{2}\frac{z+1}{z-1}.$An advantage of the Trapezoidal formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result. Of all available integration formulas, the Trapezoidal formula yields the closest match between frequency-domain properties of the discretized system and the corresponding continuous-time system. When C2 is a continuous-time controller, IFormula is ''. Default: 'ForwardEuler' DFormula Discrete integrator formula DF(z) for the derivative filter of the discrete-time pidstd2 controller C2. The relationship between the inputs and output of C2 is given by: $u={K}_{p}\left[\left(br-y\right)+\frac{1}{{T}_{i}}IF\left(z\right)\left(r-y\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\left(cr-y\right)\right].$ DFormula can take the following values: 'ForwardEuler' — DF(z) = $\frac{{T}_{s}}{z-1}.$This formula is best for small sample time, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sample time, the ForwardEuler formula can result in instability, even when discretizing a system that is stable in continuous time.'BackwardEuler' — DF(z) = $\frac{{T}_{s}z}{z-1}.$An advantage of the BackwardEuler formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result.'Trapezoidal' — DF(z) = $\frac{{T}_{s}}{2}\frac{z+1}{z-1}.$An advantage of the Trapezoidal formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result. Of all available integration formulas, the Trapezoidal formula yields the closest match between frequency-domain properties of the discretized system and the corresponding continuous-time system.The Trapezoidal value for DFormula is not available for a pidstd2 controller with no derivative filter (N = Inf). When C2 is a continuous-time controller, DFormula is ''. Default: 'ForwardEuler' InputDelay Time delay on the system input. InputDelay is always 0 for a pidstd2 controller object. OutputDelay Time delay on the system Output. OutputDelay is always 0 for a pidstd2 controller object. Ts Sample time. For continuous-time models, Ts = 0. For discrete-time models, Ts is a positive scalar representing the sampling period. This value is expressed in the unit specified by the TimeUnit property of the model. PID controller models do not support unspecified sample time (Ts = -1). Changing this property does not discretize or resample the model. Use c2d and d2c to convert between continuous- and discrete-time representations. Use d2d to change the sample time of a discrete-time system. Default: 0 (continuous time) TimeUnit Units for the time variable, the sample time Ts, and any time delays in the model, specified as one of the following values:'nanoseconds''microseconds''milliseconds''seconds' 'minutes''hours''days''weeks''months''years' Changing this property has no effect on other properties, and therefore changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior. Default: 'seconds' InputName Input channel name, specified as a character vector or a 2-by-1 cell array of character vectors. Use this property to name the input channels of the controller model. For example, assign the names setpoint and measurement to the inputs of a 2-DOF PID controller model C as follows. C.InputName = {'setpoint';'measurement'}; Alternatively, use automatic vector expansion to assign both input names. For example: C.InputName = 'C-input'; The input names automatically expand to {'C-input(1)';'C-input(2)'}. You can use the shorthand notation u to refer to the InputName property. For example, C.u is equivalent to C.InputName. Input channel names have several uses, including: Identifying channels on model display and plotsSpecifying connection points when interconnecting models Default: {'';''} InputUnit Input channel units, specified as a 2-by-1 cell array of character vectors. Use this property to track input signal units. For example, assign the units Volts to the reference input and the concentration units mol/m^3 to the measurement input of a 2-DOF PID controller model C as follows. C.InputUnit = {'Volts';'mol/m^3'}; InputUnit has no effect on system behavior. Default: {'';''} InputGroup Input channel groups. This property is not needed for PID controller models. Default: struct with no fields OutputName Output channel name, specified as a character vector. Use this property to name the output channel of the controller model. For example, assign the name control to the output of a controller model C as follows. C.OutputName = 'control'; You can use the shorthand notation y to refer to the OutputName property. For example, C.y is equivalent to C.OutputName. Input channel names have several uses, including: Identifying channels on model display and plotsSpecifying connection points when interconnecting models Default: Empty character vector, '' OutputUnit Output channel units, specified as a character vector. Use this property to track output signal units. For example, assign the unit Volts to the output of a controller model C as follows. C.OutputUnit = 'Volts'; OutputUnit has no effect on system behavior. Default: Empty character vector, '' OutputGroup Output channel groups. This property is not needed for PID controller models. Default: struct with no fields Name System name, specified as a character vector. For example, 'system_1'. Default: '' Notes Any text that you want to associate with the system, stored as a string or a cell array of character vectors. The property stores whichever data type you provide. For instance, if sys1 and sys2 are dynamic system models, you can set their Notes properties as follows: sys1.Notes = "sys1 has a string."; sys2.Notes = 'sys2 has a character vector.'; sys1.Notes sys2.Notes ans = "sys1 has a string." ans = 'sys2 has a character vector.' Default: [0×1 string] UserData Any type of data you want to associate with system, specified as any MATLAB® data type. Default: [] SamplingGrid Sampling grid for model arrays, specified as a data structure. For model arrays that are derived by sampling one or more independent variables, this property tracks the variable values associated with each model in the array. This information appears when you display or plot the model array. Use this information to trace results back to the independent variables. Set the field names of the data structure to the names of the sampling variables. Set the field values to the sampled variable values associated with each model in the array. All sampling variables should be numeric and scalar valued, and all arrays of sampled values should match the dimensions of the model array. For example, suppose you create a 11-by-1 array of linear models, sysarr, by taking snapshots of a linear time-varying system at times t = 0:10. The following code stores the time samples with the linear models. sysarr.SamplingGrid = struct('time',0:10) Similarly, suppose you create a 6-by-9 model array, M, by independently sampling two variables, zeta and w. The following code attaches the (zeta,w) values to M. [zeta,w] = ndgrid(<6 values of zeta>,<9 values of w>) M.SamplingGrid = struct('zeta',zeta,'w',w) When you display M, each entry in the array includes the corresponding zeta and w values. M M(:,:,1,1) [zeta=0.3, w=5] = 25 -------------- s^2 + 3 s + 25 M(:,:,2,1) [zeta=0.35, w=5] = 25 ---------------- s^2 + 3.5 s + 25 ... For model arrays generated by linearizing a Simulink® model at multiple parameter values or operating points, the software populates SamplingGrid automatically with the variable values that correspond to each entry in the array. For example, the Simulink Control Design™ commands linearize (Simulink Control Design) and slLinearizer (Simulink Control Design) populate SamplingGrid in this way. Default: []

## Examples

collapse all

Create a continuous-time 2-DOF PDF controller in standard form. To do so, set the integral time constant to Inf. Set the other gains and the filter divisor to the desired values.

Kp = 1;
Ti = Inf;    % No integrator
Td = 3;
N = 6;
b = 0.5;    % setpoint weight on proportional term
c = 0.5;    % setpoint weight on derivative term
C2 = pidstd2(Kp,Ti,Td,N,b,c)
C2 =

s
u = Kp * [(b*r-y) + Td * ------------ * (c*r-y)]
(Td/N)*s+1

with Kp = 1, Td = 3, N = 6, b = 0.5, c = 0.5

Continuous-time 2-DOF PDF controller in standard form

The display shows the controller type, formula, and parameter values, and verifies that the controller has no integrator term.

Create a discrete-time 2-DOF PI controller in standard form, using the trapezoidal discretization formula. Specify the formula using Name,Value syntax.

Kp = 1;
Ti = 2.4;
Td = 0;
N = Inf;
b = 0.5;
c = 0;
Ts = 0.1;
C2 = pidstd2(Kp,Ti,Td,N,b,c,Ts,'IFormula','Trapezoidal')
C2 =

1     Ts*(z+1)
u = Kp * [(b*r-y) + ---- * -------- * (r-y)]
Ti    2*(z-1)

with Kp = 1, Ti = 2.4, b = 0.5, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time 2-DOF PI controller in standard form

Setting Td = 0 specifies a PI controller with no derivative term. As the display shows, the values of N and c are not used in this controller. The display also shows that the trapezoidal formula is used for the integrator.

Create a 2-DOF PID controller in standard form, and set the dynamic system properties InputName and OutputName. Naming the inputs and the output is useful, for example, when you interconnect the PID controller with other dynamic system models using the connect command.

C2 = pidstd2(1,2,3,10,1,1,'InputName',{'r','y'},'OutputName','u')
C2 =

1      1                      s
u = Kp * [(b*r-y) + ---- * --- * (r-y) + Td * ------------ * (c*r-y)]
Ti     s                  (Td/N)*s+1

with Kp = 1, Ti = 2, Td = 3, N = 10, b = 1, c = 1

Continuous-time 2-DOF PIDF controller in standard form

A 2-DOF PID controller has two inputs and one output. Therefore, the 'InputName' property is an array containing two names, one for each input. The model display does not show the input and output names for the PID controller, but you can examine the property values to see them. For instance, verify the input name of the controller.

C2.InputName
ans = 2x1 cell
{'r'}
{'y'}

Create a 2-by-3 grid of 2-DOF PI controllers in standard form. The proportional gain ranges from 1–2 across the array rows, and the integrator time constant ranges from 5–9 across columns.

To build the array of PID controllers, start with arrays representing the gains.

Kp = [1 1 1;2 2 2];
Ti = [5:2:9;5:2:9];

When you pass these arrays to the pidstd2 command, the command returns the array of controllers.

pi_array = pidstd2(Kp,Ti,0,Inf,0.5,0,'Ts',0.1,'IFormula','BackwardEuler');
size(pi_array)
2x3 array of 2-DOF PID controller.
Each PID has 1 output and 2 inputs.

If you provide scalar values for some coefficients, pidstd2 automatically expands them and assigns the same value to all entries in the array. For instance, in this example, Td = 0, so that all entries in the array are PI controllers. Also, all entries in the array have b = 0.5.

Access entries in the array using array indexing. For dynamic system arrays, the first two dimensions are the I/O dimensions of the model, and the remaining dimensions are the array dimensions. Therefore, the following command extracts the (2,3) entry in the array.

pi23 = pi_array(:,:,2,3)
pi23 =

1      Ts*z
u = Kp * [(b*r-y) + ---- * ------ * (r-y)]
Ti      z-1

with Kp = 2, Ti = 9, b = 0.5, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time 2-DOF PI controller in standard form

You can also build an array of PID controllers using the stack command.

C2 = pidstd2(1,5,0.1,Inf,0.5,0.5);         % PID controller
C2f = pidstd2(1,5,0.1,0.5,0.5,0.5);        % PID controller with filter
pid_array = stack(2,C2,C2f);               % stack along 2nd array dimension

These commands return a 1-by-2 array of controllers.

size(pid_array)
1x2 array of 2-DOF PID controller.
Each PID has 1 output and 2 inputs.

All PID controllers in an array must have the same sample time, discrete integrator formulas, and dynamic system properties such as InputName and OutputName.

Convert a parallel-form pid2 controller to standard form.

Parallel PID form expresses the controller actions in terms of proportional, integral, and derivative gains Kp, Ki, and Kd, and filter time constant Tf. You can convert a parallel-form pid2 controller to standard form using the pidstd2 command, provided that both of the following are true:

• The pid2 controller can be expressed in valid standard form.

• The gains Kp, Ki, and Kd of the pid2 controller all have the same sign.

For example, consider the following parallel-form controller.

Kp = 2;
Ki = 3;
Kd = 4;
Tf = 2;
b = 0.1;
c = 0.5;
C2_par = pid2(Kp,Ki,Kd,Tf,b,c)
C2_par =

1                s
u = Kp (b*r-y) + Ki --- (r-y) + Kd -------- (c*r-y)
s              Tf*s+1

with Kp = 2, Ki = 3, Kd = 4, Tf = 2, b = 0.1, c = 0.5

Continuous-time 2-DOF PIDF controller in parallel form.

Convert this controller to parallel form using pidstd2.

C2_std = pidstd2(C2_par)
C2_std =

1      1                      s
u = Kp * [(b*r-y) + ---- * --- * (r-y) + Td * ------------ * (c*r-y)]
Ti     s                  (Td/N)*s+1

with Kp = 2, Ti = 0.667, Td = 2, N = 1, b = 0.1, c = 0.5

Continuous-time 2-DOF PIDF controller in standard form

The display confirms the new standard form. A response plot confirms that the two forms are equivalent.

bodeplot(C2_par,'b-',C2_std,'r--')
legend('Parallel','Standard','Location','Southeast')

Convert a two-input, one-output continuous-time dynamic system that represents a 2-DOF PID controller to a standard-form pidstd2 controller.

The following state-space matrices represent a 2-DOF PID controller.

A = [0,0;0,-8.181];
B = [1,-1;-0.1109,8.181];
C = [0.2301,10.66];
D = [0.8905,-11.79];
sys = ss(A,B,C,D);

Rewrite sys in terms of the standard-form PID parameters Kp, Ti, Td, and N, and the setpoint weights b and c.

C2 = pidstd2(sys)
C2 =

1      1                      s
u = Kp * [(b*r-y) + ---- * --- * (r-y) + Td * ------------ * (c*r-y)]
Ti     s                  (Td/N)*s+1

with Kp = 1.13, Ti = 4.91, Td = 1.15, N = 9.43, b = 0.66, c = 0.0136

Continuous-time 2-DOF PIDF controller in standard form

Convert a discrete-time dynamic system that represents a 2-DOF PID controller with derivative filter to standard pidstd2 form.

The following state-space matrices represent a discrete-time 2-DOF PID controller with a sample time of 0.05 s.

A = [1,0;0,0.6643];
B = [0.05,-0.05; -0.004553,0.3357];
C = [0.2301,10.66];
D = [0.8905,-11.79];
Ts = 0.05;
sys = ss(A,B,C,D,Ts);

When you convert sys to 2-DOF PID form, the result depends on which discrete integrator formulas you specify for the conversion. For instance, use the default, ForwardEuler, for both the integrator and the derivative.

C2fe = pidstd2(sys)
C2fe =

1       Ts                         1
u = Kp * [(b*r-y) + ---- * ------ * (r-y) + Td * --------------- * (c*r-y)]
Ti      z-1                 (Td/N)+Ts/(z-1)

with Kp = 1.13, Ti = 4.91, Td = 1.41, N = 9.43, b = 0.66, c = 0.0136, Ts = 0.05

Sample time: 0.05 seconds
Discrete-time 2-DOF PIDF controller in standard form

Now convert using the Trapezoidal formula.

C2trap = pidstd2(sys,'IFormula','Trapezoidal','DFormula','Trapezoidal')
C2trap =

1     Ts*(z+1)                           1
u = Kp * [(b*r-y) + ---- * -------- * (r-y) + Td * ----------------------- * (c*r-y)]
Ti    2*(z-1)                 (Td/N)+Ts/2*(z+1)/(z-1)

with Kp = 1.12, Ti = 4.89, Td = 1.41, N = 11.4, b = 0.658, c = 0.0136, Ts = 0.05

Sample time: 0.05 seconds
Discrete-time 2-DOF PIDF controller in standard form

The displays show the difference in resulting coefficient values and functional form.

For some dynamic systems, attempting to use the Trapezoidal or BackwardEuler integrator formulas yields invalid results, such as negative Ti, Td, or N values. In such cases, pidstd2 returns an error.

Discretize a continuous-time standard-form 2-DOF PID controller and specify the integral and derivative filter formulas.

Create a continuous-time pidstd2 controller and discretize it using the zero-order-hold method of the c2d command.

C2con = pidstd2(10,5,3,0.5,1,1);  % continuous-time 2-DOF PIDF controller
C2dis1 = c2d(C2con,0.1,'zoh')
C2dis1 =

1       Ts                         1
u = Kp * [(b*r-y) + ---- * ------ * (r-y) + Td * --------------- * (c*r-y)]
Ti      z-1                 (Td/N)+Ts/(z-1)

with Kp = 10, Ti = 5, Td = 3.03, N = 0.5, b = 1, c = 1, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time 2-DOF PIDF controller in standard form

The display shows that c2d computes new PID coefficients for the discrete-time controller.

The discrete integrator formulas of the discretized controller depend on the c2d discretization method, as described in Tips. For the zoh method, both IFormula and DFormula are ForwardEuler.

C2dis1.IFormula
ans =
'ForwardEuler'
C2dis1.DFormula
ans =
'ForwardEuler'

If you want to use different formulas from the ones returned by c2d, then you can directly set the Ts, IFormula, and DFormula properties of the controller to the desired values.

C2dis2 = C2con;
C2dis2.Ts = 0.1;
C2dis2.IFormula = 'BackwardEuler';
C2dis2.DFormula = 'BackwardEuler';

However, these commands do not compute new coefficients for the discretized controller. To see this, examine C2dis2 and compare the coefficients to C2con and C2dis1.

C2dis2
C2dis2 =

1      Ts*z                         1
u = Kp * [(b*r-y) + ---- * ------ * (r-y) + Td * ----------------- * (c*r-y)]
Ti      z-1                 (Td/N)+Ts*z/(z-1)

with Kp = 10, Ti = 5, Td = 3, N = 0.5, b = 1, c = 1, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time 2-DOF PIDF controller in standard form

## Tips

• To design a PID controller for a particular plant, use pidtune or pidTuner. To create a tunable 2-DOF PID controller as a control design block, use tunablePID2.

• To break a 2-DOF controller into two SISO control components, such as a feedback controller and a feedforward controller, use getComponents.

• Create arrays of pidstd2 controllers by:

• Specifying array values for one or more of the coefficients Kp, Ti, Td, N, b, and c.

• Specifying an array of dynamic systems sys to convert to pid2 controller objects.

• Using stack to build arrays from individual controllers or smaller arrays.

• Passing an array of plant models to pidtune.

In an array of pidstd2 controllers, each controller must have the same sample time Ts and discrete integrator formulas IFormula and DFormula.

• To create or convert to a parallel-form controller, use pid2. Parallel form expresses the controller actions in terms of proportional, integral, and derivative gains Kp, Ki and Kd, and a filter time constant Tf. For example, the relationship between the inputs and output of a continuous-time parallel-form 2-DOF PID controller is given by:

$u={K}_{p}\left(br-y\right)+\frac{{K}_{i}}{s}\left(r-y\right)+\frac{{K}_{d}s}{{T}_{f}s+1}\left(cr-y\right).$

• There are two ways to discretize a continuous-time pidstd2 controller:

• Use the c2d command. c2d computes new parameter values for the discretized controller. The discrete integrator formulas of the discretized controller depend upon the c2d discretization method you use, as shown in the following table.

'zoh'ForwardEulerForwardEuler
'foh'TrapezoidalTrapezoidal
'tustin'TrapezoidalTrapezoidal
'impulse'ForwardEulerForwardEuler
'matched'ForwardEulerForwardEuler

• If you require different discrete integrator formulas, you can discretize the controller by directly setting Ts, IFormula, and DFormula to the desired values. (See Discretize a Standard-Form 2-DOF PID Controller.) However, this method does not compute new gain and filter-constant values for the discretized controller. Therefore, this method might yield a poorer match between the continuous- and discrete-time pidstd2 controllers than using c2d.